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ECE317 : Feedback and Control
Lecture:ODE solution via Laplace transform
Dr. Richard Tymerski
Dept. of Electrical and Computer Engineering
Portland State University
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Course roadmap
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Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
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Laplace transform (review)
• One of most important math tools in the course!
• Definition: For a function f(t) (f(t)=0 for t<0),
• We denote Laplace transform of f(t) by F(s).
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f(t)
t0
F(s)
(s: complex variable)A:=B (A is defined by B.)
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Laplace transform table
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Inverse Laplace
Transform
(u(t) is often omitted.)
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Advantages of s-domain (review)
• We can transform an ordinary differential equation into an algebraic equation which is easy to solve.
(This lecture)
• It is easy to analyze and design interconnected (series, feedback etc.) systems.
• Frequency domain information of signals can be easily dealt with.
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An advantage of Laplace transform
• We can transform an ordinary differential equation (ODE) into an algebraic equation (AE).
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ODE AE
Partial fraction
expansionSolution to ODE
t-domain s-domain
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Example 1 (distinct roots)
ODE with initial conditions (ICs)
1. Laplace transform
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distinct roots
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Properties of Laplace transformDifferentiation (review)
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t-domain
s-domain
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Example 1 (cont’d)
2. Partial fraction expansion
Multiply both sides by s(s+1)(s+2) :
Compare coefficients:
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unknowns
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Example 1 (cont’d)
3. Inverse Laplace transform
If we are interested in only the final value of y(t), apply the Final Value Theorem, without explicitly computing y(t):
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(You may omit u(t).)
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Example 2 (repeated roots)
ODE with zero initial conditions (ICs)
1. Laplace transform
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Repeated roots
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Example 2 (cont’d)
2. Partial fraction expansion
Multiply both sides by
Compare coefficients:
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unknowns
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Example 2 (cont’d)
3. Inverse Laplace transform
If we are interested in only the final value of y(t), apply the Final Value Theorem, without explicitly computing y(t):
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(u(t) omitted.)
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Properties of Laplace transformFrequency shift theorem (review)
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Proof.t-domain
s-domain
Ex.
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Example 3 (complex roots)
ODE with zero initial conditions (ICs)
1. Laplace transform
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Complex roots
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Example 3 (cont’d)
2. Partial fraction expansion
Multiply both sides by
Compare coefficients:
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unknowns
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Example 3 (cont’d)
3. Inverse Laplace transform
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Laplace transform table
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Frequency shift theorem
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Summary
• Solution to ODE via Laplace transform1. Laplace transform
2. Partial fraction expansion
3. Inverse Laplace transform
• Next, modeling of physical systems in s-domain
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